"978-80-7378-172-9" . "Pap\u00EDrov\u00E1 geometrie v dev\u00EDti jedn\u00E1n\u00EDch" . . "Pap\u00EDrov\u00E1 geometrie v dev\u00EDti jedn\u00E1n\u00EDch"@cs . . . . "Praha" . "22"^^ . "Fiala, Ji\u0159\u00ED" . "P(GAP401/10/0690)" . "Pap\u00EDrov\u00E1 geometrie v dev\u00EDti jedn\u00E1n\u00EDch" . "1"^^ . . "1"^^ . "219477" . "2011-08-26+02:00"^^ . "MATFYZPRESS" . "1. Co lze skl\u00E1d\u00E1n\u00EDm pap\u00EDru z\u00EDskat? Pokud se vezme nestandardn\u00ED skl\u00E1d\u00E1n\u00ED, pak dokonce trisekce \u00FAhlu. 2. Standardn\u00ED pap\u00EDrov\u00E9 skl\u00E1d\u00E1n\u00ED je ekvivalentn\u00ED geometrii, ve kter\u00E9 m\u016F\u017Eeme spojit dva dan\u00E9 body p\u0159\u00EDmkou a posouvat j\u00ED. 3. Takov\u00E1to geometrie m\u00E1 model, kter\u00FD spl\u0148uje v\u0161echny Hilbertovy axiomy a\u017E na axiom \u00FAplnosti. 4. Co lze zkonstruovat v takov\u00E9to geometrii? Pouze tot\u00E1ln\u00ED re\u00E1ln\u00E1 \u010D\u00EDsla (jak odpov\u00EDd\u00E1 Hilbert). 5. Jak je lze zkonstruovat: to je to, co cht\u011Bl Hilbert poznat, ale nepovedlo se mu to, tak\u017Ee formulovat zn\u00E1m\u00FD 17. probl\u00E9m. 6. Tento probl\u00E9m po t\u00E9m\u011B\u0159 t\u0159iceti letech vy\u0159e\u0161ili E. Artin a O. Schreier (k tomu vytvo\u0159il kr\u00E1snou teorii re\u00E1ln\u00FDch pol\u00ED). 7. Bohu\u017Eel toto \u0159e\u0161en\u00ED nen\u00ED konstruktivn\u00ED; po skoro t\u0159iceti letech Abraham Robinson a Georg Kreisel na\u0161li konstruk\u010Dn\u00ED \u0159e\u0161en\u00ED. 8. Bohu\u017Eel odpov\u00EDdaj\u00EDc\u00ED algoritmus m\u00E1 exp exp slo\u017Eitost, tak\u017Ee je ve skute\u010Dnosti nepou\u017Eiteln\u00FD. 9. Tak\u017Ee z\u00E1v\u011Bre\u010Dn\u00E1 ot\u00E1zka je: kdy m\u016F\u017Eeme doopravdy (a nejen konvencemi) \u0159\u00EDci, \u017Ee matematick\u00FD probl\u00E9m byl definitivn\u011B vy\u0159e\u0161en?"@cs . . "Paper geometry in nine acts"@en . "1. Co lze skl\u00E1d\u00E1n\u00EDm pap\u00EDru z\u00EDskat? Pokud se vezme nestandardn\u00ED skl\u00E1d\u00E1n\u00ED, pak dokonce trisekce \u00FAhlu. 2. Standardn\u00ED pap\u00EDrov\u00E9 skl\u00E1d\u00E1n\u00ED je ekvivalentn\u00ED geometrii, ve kter\u00E9 m\u016F\u017Eeme spojit dva dan\u00E9 body p\u0159\u00EDmkou a posouvat j\u00ED. 3. Takov\u00E1to geometrie m\u00E1 model, kter\u00FD spl\u0148uje v\u0161echny Hilbertovy axiomy a\u017E na axiom \u00FAplnosti. 4. Co lze zkonstruovat v takov\u00E9to geometrii? Pouze tot\u00E1ln\u00ED re\u00E1ln\u00E1 \u010D\u00EDsla (jak odpov\u00EDd\u00E1 Hilbert). 5. Jak je lze zkonstruovat: to je to, co cht\u011Bl Hilbert poznat, ale nepovedlo se mu to, tak\u017Ee formulovat zn\u00E1m\u00FD 17. probl\u00E9m. 6. Tento probl\u00E9m po t\u00E9m\u011B\u0159 t\u0159iceti letech vy\u0159e\u0161ili E. Artin a O. Schreier (k tomu vytvo\u0159il kr\u00E1snou teorii re\u00E1ln\u00FDch pol\u00ED). 7. Bohu\u017Eel toto \u0159e\u0161en\u00ED nen\u00ED konstruktivn\u00ED; po skoro t\u0159iceti letech Abraham Robinson a Georg Kreisel na\u0161li konstruk\u010Dn\u00ED \u0159e\u0161en\u00ED. 8. Bohu\u017Eel odpov\u00EDdaj\u00EDc\u00ED algoritmus m\u00E1 exp exp slo\u017Eitost, tak\u017Ee je ve skute\u010Dnosti nepou\u017Eiteln\u00FD. 9. Tak\u017Ee z\u00E1v\u011Bre\u010Dn\u00E1 ot\u00E1zka je: kdy m\u016F\u017Eeme doopravdy (a nejen konvencemi) \u0159\u00EDci, \u017Ee matematick\u00FD probl\u00E9m byl definitivn\u011B vy\u0159e\u0161en?" . . . . "RIV/49777513:23330/11:43897215" . "23330" . "Jev\u00ED\u010Dko" . "32. mezin\u00E1rodn\u00ED konference Historie matematiky" . "1. What can be done by paper folding? If one uses non-standard folding then even the trisection of an angle. 2. Standard paper-folding is equivalent to the geometry, in which we can join two given points by a straight line and more a given straight line (%22standard%22) to a given place. 3. Such a geometry forms a model, which satisfies all the Hilbert axioms except the axiom of completeness. 4. What can be constructed in such geometry? Only totally real numbers. 5. How it can be constructed: that is what Hilbert wanted to know, but was unable to do, so he made of it he famous 17th problem. 6. This problem was solved by E. Artin and O. Schreier in 1930s. 7. Unfortunately this solution is not constructive; in 1960s a constructive solution was given by Abraham Robinson and Georg Kreisel. 8. Unfortunately corresponding algorithm has exp exp complexity, so it is in fact inapplicable. 9. So the final question is: when we can really (not by convention) say that a mathematical problem was definitely solved?"@en . . "RIV/49777513:23330/11:43897215!RIV12-GA0-23330___" . . "[0E7A9A51EF2D]" . . . "solvability of mathematical problems; 17th Hilbert problem; paper folding; geometry; history of 20th centrury mathematics"@en . . "Pap\u00EDrov\u00E1 geometrie v dev\u00EDti jedn\u00E1n\u00EDch"@cs . . . "Paper geometry in nine acts"@en . . . .